Make the presentation in terms of likelihood ratios rather than probabilities. Then Bayes becomes plain multiplication. Multiple hypotheses and multiple pieces of evidence are also easier to handle in this format.

Since thinking of Bayes like this, I'm more likely to do simple calculations in my head and I have a better handle on the proper size of an update. Previously, it didn't click that going from 0.5 to 0.75 takes the same amount of evidence as 0.75 to 0.9 and 0.9 to 0.96. In likelihood terms, that is going from 1:1 to 3:1 to 9:1 to 27:1, with an additional 3:1 piece of evidence at each stage. Eliezer is rewriting An Intuitive Explanation with likelihood ratios as well.

This is my favorite explanation. It's short and to the point, and the visuals are excellent.

I like the Intuitive Explanation of Intuitive Explanation of Bayes' Theorem a lot, but it might be too simple for programmers and natural science students.

Have you looked at Eliezer's intuitive explanation? I think a verbalized version of that might be effective.